Magnetoresistance effects and magnetoelastic behavior of ferromagnetic materials 

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Basic notions of material behavior

AMR effect is closely related to the magnetic behavior of ferromagnetic materials. This work will investigate the role of mechanical stress on this behavior. This section gives a brief overview on the notions of material behavior used in the document.

Different types of magnetic behavior

Materials may be classified by their response to externally applied magnetic fields as diamagnetic, paramagnetic, or ferromagnetic. These magnetic responses differ greatly in strength [Bozorth, 1993]. Diamagnetism is a property of all materials and opposes applied magnetic fields, but is very weak. Paramagnetism, when present, is stronger than diamagnetism and produces magnetization in the direction of the applied field, and proportional to the applied field. Ferromagnetic effects are very large, producing magnetization sometimes orders of magnitude greater than the applied field and as such are much larger than either diamagnetic or paramagnetic effects.
The magnetization of a material is expressed in terms of density of net magnetic dipole moments ~m in the material. We define a vector quantity called the magnetization ( ~M ) by ~M = μt~otal/V.

Magnetoelastic coupling

Magnetostriction is the deformation of a magnetic material due to magnetic interactions. When iron is cooled down from a high temperature through its Curie temperature, an anomalous isotropic expansion is observed near the Curie temperature.
This slightly magnetic field-dependent anomaly associated with the magnetism of iron (and other magnetic substances) is called volume magnetostriction. This is the isotropic aspect of the spontaneous magnetostriction. Now, if a magnetic field is applied to the iron sample, an additional anisotropic deformation that stretches or shrinks the sample in the direction of the magnetic field is observed. This field-dependent phenomenon is called Joule magnetostriction; it is the anisotropic aspect of the forced magnetostriction.

AMR models in the literature

Several technologies can be used to ensure an optimal performance of AMR sensors [Tumanski, 2001, Stutzke et al., 2005, Kubik et al., 2006, Zimmermann et al., 2005], but this performance is always related to the intricate relationship between the microstructure and the macroscopic response of the sensor. In the case of ferromagnetic polycrystalline materials, the microstructure scale is twofold. The material is divided into magnetic domains with different magnetization, and it is also divided into grains with different crystallographic orientations. From the modeling point of view, a large part of phenomenological models for the AMR effect are based on the hypothesis that the material is at magnetic saturation so that it can be described as a one-domain particle [McGuire and Potter, 1975, Li et al., 2010, Beltran et al., 2007]. This is however a strong simplification given the complex magnetic domain structure of ferromagnetic materials.
Some authors proposed micromagnetic calculations to describe this evolving domain structure in thin film AMR sensors [Koehler et al., 1993, Shiiki et al., 1996]. But these approaches usually lead to dissuasive computation time so that strong simplifying assumptions are needed when describing real systems.
An alternative is the use of a micro-macro approach that incorporates a statistical view of the microstructure to define the effective properties of ferromagnetic materials [Daniel et al., 2008, Daniel and Galopin, 2008]. In this thesis this micromacro – or multiscale – approach is used to describe the properties of thin film AMR sensors. This model allows us reproducing the main features of AMR magnetic field sensors under typical operating conditions.

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Magnetocrystalline anisotropy energy

Magnetocrystalline anisotropy energy is the energy necessary to deflect the magnetic moment in a single crystal from the easy to the hard direction. The easy and hard directions arise from the interaction of the spin magnetic moment with the crystal lattice (spin-orbit coupling). The magnetocrystalline energy is minimal when the magnetization is aligned with an easy axis and maximal when it is aligned with a hard axis of the single crystal.
The magnetocrystalline anisotropy energy is generally represented as a spherical expansion in powers of the direction cosines of the magnetization. In cubic crystals (like Iron or Nickel) it is sufficient to represent the anisotropy energy in an arbitrary direction by just the first two terms in the series expansion. These two terms each have an empirical constant associated with them called the first- and second order anisotropy constants, K1 and K2 respectively.
In the case of cubic crystallographic structure the magnetocrystalline energy can be written [Cullity and Graham, 2011]: WK = K1(2 12 2 + 2 22 3 + 2 32 1) + K2(2 12 22 3).

Calculation at the magnetic domain scale

The lowest calculation scale in our model is the magnetic domain scale. A magnetic domain is a region in which the magnetization vectors are aligned. The magnetization is then uniform in a domain, noted , and given by M = Ms with Ms the saturation magnetization of the material and = t[1 2 3] the direction cosines of the magnetization. In the following we take the definition of the potential energy given by equation (2.1) and try to make some simplifications associated to the uniformity of the magnetization and the uniformity of elastic coefficients within the domain.

Table of contents :

Introduction
1 Magnetoresistance effects and magnetoelastic behavior of ferromagnetic materials 
1.1 Magnetoresistive elements for magnetic field sensing
1.1.1 Hall sensors
1.1.2 Anisotropic Magnetoresistance (AMR) based sensors
1.1.3 Giant, Colossal and Tunnel Magnetoresistance (GMR, CMR and TMR) based sensors
1.1.4 Comparison with other magnetoresistive elements
1.2 Basic notions of material behavior
1.2.1 Different types of magnetic behavior
1.2.2 Ferromagnetism
1.2.3 Mechanical behavior
1.2.4 Magnetoelastic coupling
1.3 AMR models in the literature
2 Single crystal behavior 
2.1 Microscopic magneto-elastic model
2.1.1 Exchange energy
2.1.2 Magnetocrystalline anisotropy energy
2.1.3 Magnetostatic energy
2.1.4 Elastic energy
2.2 Calculation at the magnetic domain scale
2.2.1 Potential energy of a magnetic domain
2.2.2 Single domain model of AMR
2.3 Calculation at the single crystal scale
2.3.1 Selection and calculation of state variables
2.3.2 Homogenization
2.4 Results and comparison to experimental data
2.4.1 Magnetoelastic properties
2.4.2 Magnetoresistive properties
2.5 Conclusion
3 Polycrystal behavior 
3.1 Macroscopic model
3.1.1 Modeling strategy
3.1.2 Localization step
3.1.3 Calculation of the effective properties
3.1.4 Calculation algorithm and model parameters
3.2 Prediction of the AMR effect on ferromagnetic polycrystals
3.2.1 Effect of stress on the magnetoresistive behavior
3.2.2 Effect of crystallographic texture
3.3 Conclusion
4 Modeling of thin film AMR sensor properties 
4.1 Design and contruction of AMR sensors
4.2 Modeling thin film properties
4.2.1 Introduction of surface effect
4.2.2 Textured AMR thin film sensor properties
4.3 Effect of biasing magnetic field
4.3.1 Definition of the sensitivity of an AMR sensor element .
4.4 Influence of the film thickness
4.5 Effect of stress on the properties of AMR thin film sensors
4.6 Conclusion
General conclusion
Bibliography

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